The following computer-generated description may contain errors and does not represent the quality of the book:Oct. more from its original scope and purpose, threatening to sacrifice its earlier unity and to split into diverse branches. At the same time the attention bestowed upon it by the general scientific public diminished in equal proportions. It became almost the custom to regard modern mathematical speculation as something having no general interest or importance; and the proposal was even made that, at least for purposes of instruction, all results be formulated from the same standpoints as in the earlier period. Such conditions were unquestionably to be regretted. This is a picture of the past. I wish on the present occasion to state and to emphasize the fact that in the last two decades a marked improvement from within has asserted itself in our science, with constantly increasing success. The matter has been found simpler than was at first believed. It appears, indeed, that the different branches of mathematics have actually developed not in diverging but in parallel directions; that it is possible to combine their results into certain general conceptions. Such a conception is that of the function in particular that of the analytical function of the complex variable. Another conception of perhaps the same range is that of the group which just now stands in the foreground of mathematical progress. Proceeding from the idea of groups, we learn more and more to co-ordinate and connect different mathematical sciences. Thus, for example, geometry and the theory of numbers, which long seemed to represent antagonistic tendencies, no longer form an antithesis, but have come in many ways to appear as different aspects of one and the same theory. This unifying tendency, originally purely theoretical, must inevitably extend to the applications of mathematics in other sciences, and on the other hand is sustained and reinforced in the development and extension of tliese latter. I presume that specific examples of this interchange of influence between pure and applied mathematics may be not without interest for the members of this genei al session, and on this account have selected for brief preliminary mention two of the papers which I have later to present to the mathematical section. The first of these papers (by Dr. Schonflies) presents a review of the progress of mathematical crystallography. Sohncke, about 1877, treated ciystals as aggregates of congruent molecules of any shape whatever, regularly arranged in space. In 1884 Fedorow made further progress by admitting the hypothesis that the molecules might be in part inversely instead of directly congruent.